We introduce a notion of expected utility for quantum tasks and discuss some general conditions under which this is increased by the presence of quantum noise in the underlying resource states. We apply the resulting formalism to the specific problem of playing the parity game with ground states of the random transverse-field Ising model. This demonstrates a separation in the ground-state phase diagram between regions where rational players will be ``risk-seeking'' or ``risk-averse'', depending on whether they win the game more or less often in the presence of disorder. The boundary between these regions depends non-universally on the correlation length of the disorder. Strikingly, we find that adding zero-mean, uncorrelated disorder to the transverse fields can generate a weak quantum advantage that would not exist in the absence of noise.
We introduce a notion of expected utility for quantum tasks and discuss some general conditions under which this is increased by the presence of quantum noise in the underlying resource states. We apply the resulting formalism to the specific problem of playing the parity game with ground states of the random transverse-field Ising model. This demonstrates a separation in the ground-state phase diagram between regions where rational players will be risk-seeking'' or risk-averse'', depending on whether they win the game more or less often in the presence of disorder. The boundary between these regions depends non-universally on the correlation length of the disorder. Strikingly, we find that adding zero-mean, uncorrelated disorder to the transverse fields can generate a weak quantum advantage that would not exist in the absence of noise.
Sharing correlated random variables is a resource for a number of information theoretic tasks such as privacy amplification, simultaneous message passing, secret sharing and many more. In this article, we show that to establish such a resource called shared randomness, quantum systems provide an advantage over their classical counterpart. Precisely, we show that appropriate albeit fixed measurements on a shared two-qubit state can generate correlations which cannot be obtained from any possible state on two classical bits. In a resource theoretic set-up, this feature of quantum systems can be interpreted as an advantage in winning a two players co-operative game, which we call the `non-monopolize social subsidy' game. It turns out that the quantum states leading to the desired advantage must possess non-classicality in the form of quantum discord. On the other hand, while distributing such sources of shared randomness between two parties via noisy channels, quantum channels with zero capacity as well as with classical capacity strictly less than unity perform more efficiently than the perfect classical channel. Protocols presented here are noise-robust and hence should be realizable with state-of-the-art quantum devices.
The last two decades have witnessed a rapid development of quantum information processing, a new paradigm which studies the power and limit of "quantum advantages" in various information processing tasks. Problems such as when quantum advantage exists, and if existing, how much it could be, are at a central position of these studies. In a broad class of scenarios, there are, implicitly or explicitly, at least two parties involved, who share a state, and the correlation in this shared state is the key factor to the efficiency under concern. In these scenarios, the shared \emph{entanglement} or \emph{discord} is usually what accounts for quantum advantage. In this paper, we examine a fundamental problem of this nature from the perspective of game theory, a branch of applied mathematics studying selfish behaviors of two or more players. We exhibit a natural zero-sum game, in which the chance for any player to win the game depends only on the ending correlation. We show that in a certain classical equilibrium, a situation in which no player can further increase her payoff by any local classical operation, whoever first uses a quantum computer has a big advantage over its classical opponent. The equilibrium is fair to both players and, as a shared correlation, it does not contain any discord, yet a quantum advantage still exists. This indicates that at least in game theory, the previous notion of discord as a measure of non-classical correlation needs to be reexamined, when there are two players with different objectives.
Quantum game theory investigates the influence of quantum resources on strategic decision-making. In this work, two-player quantum games based on the Transverse Field Ising Model(TFIM) are investigated under amplitude-damping decoherence. The TFIM Hamiltonian naturally produces a family of entangling gates, enabling a physically motivated implementation of quantum games. The effects of noise on Nash equilibria, players' payoffs, concurrence and coherence of the quantum states for different initial states and strategy pairs are analyzed. The results show that decoherence progressively suppresses quantum strategic advantages, with maximum damping driving all outcomes to identical classical payoffs. The concurrence and coherence analysis of the states generated in the quantum game reveal initial state and strategy dependent quantum correlation dynamics, including entanglement sudden birth and death.
Nonlocal quantum games provide proof of principle that quantum resources can confer advantage at certain tasks. They also provide a compelling way to explore the computational utility of phases of matter on quantum hardware. In a recent manuscript [Hart et al., arXiv:2403.04829] we demonstrated that a toric code resource state conferred advantage at a certain nonlocal game, which remained robust to small deformations of the resource state. In this manuscript we demonstrate that this robust advantage is a generic property of resource states drawn from topological or fracton ordered phases of quantum matter. To this end, we illustrate how several other states from paradigmatic topological and fracton ordered phases can function as resources for suitably defined nonlocal games, notably the three-dimensional toric-code phase, the X-cube fracton phase, and the double-semion phase. The key in every case is to design a nonlocal game that harnesses the characteristic braiding processes of a quantum phase as a source of contextuality. We unify the strategies that take advantage of mutual statistics by relating the operators to be measured to order and disorder parameters of an underlying generalized symmetry-breaking phase transition. Finally, we massively generalize the family of games that admit perfect strategies when codewords of homological quantum error-correcting codes are used as resources.
We analyse quantum games with correlated noise through a generalized quantization scheme. Four different combinations on the basis of entanglement of initial quantum state and the measurement basis are analysed. It is shown that the quantum player only enjoys an advantage over the classical player when both the initial quantum state and the measurement basis are in entangled form. Furthermore, it is shown that for maximum correlation the effects of decoherence diminish and it behaves as a noiseless game.
The parity game is an example of a nonlocal game: By sharing a Greenberger-Horne-Zeilinger (GHZ) state before playing this game, the players can win with a higher probability than is allowed by classical physics. The GHZ state of N qubits is also the ground state of the ferromagnetic quantum Ising model on N qubits in the limit of vanishingly weak quantum fluctuations. Motivated by this observation, we examine the probability that N players who share the ground state of a generic quantum Ising model, which exhibits nonvanishing quantum fluctuations, still win the parity game using the protocol optimized for the GHZ state. Our main result is a modified parity game for which this protocol asymptotically exhibits quantum advantage in precisely the ferromagnetic phase of the quantum Ising model. We further prove that the ground state of the exactly soluble d=1+1 transverse-field Ising model can provide a quantum advantage for the parity game over an even wider region, which includes the entire ferromagnetic phase, the critical point and part of the paramagnetic phase. By contrast, we find examples of topological phases and symmetry-protected topological (SPT) phases of matter, namely the deconfined phase of the toric code Hamiltonian and the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ SPT phase in one dimension, that do not exhibit an analogous quantum advantage away from their fixed points.